A characterization of snowflakes via rectifiability
Emanuele Caputo, Nicola Cavallucci
TL;DR
This work removes compactness, doubling, and embeddability constraints from Tyson–Wu's snowflake characterization, proving that a metric space $(X,d)$ is biLipschitz to a snowflake if and only if all its ultralimits lack nontrivial rectifiable content, equivalently are purely $1$-unrectifiable and carry no nontrivial metric $1$-currents with inner-regular mass. The authors develop a general framework using ultralimits and non-standard analysis, including SRA$_k(\varepsilon,\alpha)$–based small rough-angle conditions and a line-fitting obstruction, to establish the equivalences (i)–(v) and their biLipschitz invariance. They also connect these geometric criteria to the vanishing of metric currents $\oldsymbol{M}_1^{reg}$ and $\oldsymbol{N}_1^{reg}$ on ω-limits, providing new characterizations via geometric measure theory in metric spaces. The paper then demonstrates applications: resolving issues around non-doubling examples, deriving product stability for snowflake properties, and outlining implications for quasi-selfsimilar spaces and higher-dimensional generalizations. Overall, the work deepens the bridge between fractal geometry, ultralimit techniques, and metric-geometry currents, offering robust criteria for snowflake embeddability with broad potential extensions.
Abstract
We prove a generalization of Tyson-Wu's characterization of metric spaces biLipschitz equivalent to snowflakes to every metric space, by removing compactness, doubling and embeddability assumptions. We also characterize metric spaces that are biLipschitz equivalent to a snowflake in terms of the absence of non-trivial metric $1$-currents in every ultralimit, or equivalently in terms of purely $1$-unrectifiability of every ultralimit. Finally, we discuss some applications and examples.